Sum of Squares Based Nonlinear Control Design Techniques

Abstract

Many practical control systems have inherent nonlinear characteristics that cannot be ignored. However, the analysis and control of nonlinear systems are among the most challenging problems in systems and control theory. In this dissertation, we focus on Sum of Squares (SOS)-based nonlinear control designs for a class of nonlinear systems with polynomial vector fields to achieve stabilization and optimal control criteria.

SOS is a powerful and promising technique which has been widely used in recent years. It provides an efficient way for researchers to explore polynomial nonlinear systems, which can be used to approximate many nonlinear systems. The SOS decomposition has been used for nonlinear system analysis successfully; however, its potential for control system synthesis and design has not been fully explored. Therefore, in this research, the usage of newly emerging SOS decomposition for optimal and adaptive controller design is a unique feature, which has not been addressed in the control literature to date.

In the first part of this dissertation, we study the stabilization of state-dependent/output-dependent polynomial nonlinear systems using rational Lyapunov Functions. A direct consequence of this approach is that the Lyapunov functions can be generalized to pre-specified forms. Therefore, compared to quadratic Lyapunov function, it provides great potential to find Lyapnov certificates through systematic SOS programming and expands the domains of attraction for nonlinear systems. This has been demonstrated on a controlled Van der Pol nonlinear ocillator.

Beyond stabilization problems, we further investigate the performance improvements of nonlinear control systems in terms of H-infinity control and adaptive control. As is known, the nonlinear H-infinity control problem is formulated as Hamilton-Jacobi-Isaacs (HJI) equations/inequalities. However, their solutions are extremely difficult not only to compute but also to represent. In the second part, we relax the stringent orthonormal and decoupling assumptions on the plant structure and develop the generalized HJI inequalities, then we focus on materializing H-infinity theory into an algorithmic procedure for polynomial nonlinear systems. Based on the idea of SOS programming, the nonlinear H-infinity control problem can be solved by reformulating HJI inequalities into convex optimization conditions. Direct applications include high-precision spacecraft attitude regulation and command tracking problems.

Finally, we study SOS-based Lyapunov redesign of adaptive controllers for uncertain polynomial nonlinear systems with matching conditions. By developing adaptive control schemes to attenuate the effects of unknown parameters on the controlled output, the Lyapunov functions are extended from quadratic to higher order and the resulting control gains are generalized from constant to parameter dependent. The synthesis conditions could be formulated as polynomial matrix inequalities and are thus solvable by recasting the resulting conditions into an SOS optimization problem. Resorting to an iterative algorithm and vector projection technique, parallel results are also developed for the un-matched parameter uncertainty case. It turns out that for both cases, the constructed adaptive control law as well as the parameter adaptation law guarantee that the states of uncertain systems are finally stabilized to some desired positions and the parameter estimation error converges to zero.